Unusually strong high-excitation C I has been detected in eleven lines of sight through the Vela supernova remnant by means of UV absorption-line studies of IUE data. Most of these lines of sight lie near the western edge of the X-ray bright region of the supernova remnant in a spatially distinct band approximately 1deg by 4deg oriented approximately north/south. The high-excitation C I (denoted C I*) is interpreted as evidence of a complex of shocked dense clouds inside the supernova remnant, due to the high pressures indicated in this region. To further analyze the properties of this region of C I*, we present new HIRES-processed IRAS data of the entire Vela SNR. A temperature map calculated from the HIRES IRAS data, based on a two-component dust model, reveals the signature of hot dust at several locations in the SNR. The hot dust is anti-correlated spatially with X-ray emission as revealed by ROSAT, as would be expected for a dusty medium interacting with a shock wave. The regions of hot dust are strongly correlated with optical filaments, supporting a scenario of dense clouds interior to the SNR that have been shocked and are now cooling behind the supernova blast wave. With few exceptions, the lines of sight to the strong C I* pass through regions of hot dust and optical filaments. Possible mechanisms for the production of the anomalously large columns of C I and C I* are discussed. Dense clouds on the back western hemisphere of the remnant may explain the relatively low X-ray emission in the western portion of the Vela supernova remnant due to the slower forward shock velocity in regions where the shock has encountered the dense clouds. An alternate explanation for the presence of neutral, excited state, and ionized species along the same line of sight may be a magnetic precusor that heats and compresses the gas ahead of the shock.